On the 2-linearity of the free group

Licata, Anthony

doi:10.48550/arxiv.1606.06444

Cited by 1 publication

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“…We will use two such lemmas -one for each kind of Garside structure -in establishing the faithfulness of the action of B W on K; we record these lemmas here. A ping pong proof of the faithfulness of a categorical action of the free group is appears in work of the first author, [Lic16].…”

Section: Monoids Andmentioning

confidence: 99%

“…However, when the Coxeter group W is finite, faithfulness is known: this was proven in type A 2 by Rouquier-Zimmermann [RZ03], in type A n by Khovanov-Seidel [KS02], in type ADE by Brav-Thomas [BT11], and for arbitrary finite W by Jensen [Jen15]. Faithfulness is also known in affine type A by work of Riche [Ric08], Ishii-Ueda-Uehara [IUU10] and Gadbled-Thiel-Wagner [GTW15], as well as in the case of the free group by work of the first author [Lic16]. Thus, at least in these cases, one can attempt to study B W via its action on the triangulated category K somewhat analogously to the way one studies W via its action on the root lattice Λ.…”

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confidence: 95%

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Braid groups of type ADE, Garside monoids, and the categorified root lattice

Licata¹,

Queffelec²

2017

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We study Artin-Tits braid groups B W of type ADE via the action of B W on the homotopy category K of graded projective zigzag modules (which categorifies the action of the Weyl group W on the root lattice). Following Brav-Thomas [BT11], we define a metric on B W induced by the canonical t-structure on K, and prove that this metric on B W agrees with the wordlength metric in the canonical generators of the standard positive monoid B + W of the braid group. We also define, for each choice of a Coxeter element c in W , a baric structure on K. We use these baric structures to define metrics on the braid group, and we identify these metrics with the word-length metrics in the Birman-Ko-Lee/Bessis dual generators of the associated dual positive monoid B ∨ W.c . As consequences, we give new proofs that the standard and dual positive monoids inject into the group, give linear-algebraic solutions to the membership problem in the standard and dual positive monoids, and provide new proofs of the faithfulness of the action of B W on K. Finally, we use the compatibility of the baric and t-structures on K to prove a conjecture of Digne and Gobet regarding the canonical word-length of the dual simple generators of ADE braid groups.Contents 10 3. Categorical ping pong 17 4. Homological interpretations of word-length metrics 28 5. Standard and dual word-length metrics and a conjecture of Digne-Gobet 37 References 391. Artin-Tits braid groups in type ADE

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Section: Monoids Andmentioning

confidence: 99%